Average Error: 52.7 → 0.2
Time: 44.1s
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c \cdot -4}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} + {\left(\left(a \cdot c\right) \cdot -4\right)}^{3}}}{\sqrt{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot -4\right)\right) + \left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\left(a \cdot c\right) \cdot -4\right)}} + b}}{2}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 52.7

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Initial simplification52.7

    \[\leadsto \frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\]
  3. Using strategy rm

  4. Applied flip--52.8

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}{a}}{2}\]

  5. Applied associate-/l/52.8

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{a \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}}{2}\]

  6. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot -4\right) \cdot a}}{a \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}{2}\]
  7. Using strategy rm

  8. Applied associate-/l*0.2

    \[\leadsto \frac{\color{blue}{\frac{c \cdot -4}{\frac{a \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a}}}}{2}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity0.2

    \[\leadsto \frac{\frac{c \cdot -4}{\color{blue}{1 \cdot \frac{a \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a}}}}{2}\]

  11. Applied associate-/r*0.2

    \[\leadsto \frac{\color{blue}{\frac{\frac{c \cdot -4}{1}}{\frac{a \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a}}}}{2}\]

  12. Simplified0.1

    \[\leadsto \frac{\frac{\frac{c \cdot -4}{1}}{\color{blue}{\sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b} + b}}}{2}\]
  13. Using strategy rm

  14. Applied flip3-+0.1

    \[\leadsto \frac{\frac{\frac{c \cdot -4}{1}}{\sqrt{\color{blue}{\frac{{\left(-4 \cdot \left(c \cdot a\right)\right)}^{3} + {\left(b \cdot b\right)}^{3}}{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}}} + b}}{2}\]

  15. Applied sqrt-div0.2

    \[\leadsto \frac{\frac{\frac{c \cdot -4}{1}}{\color{blue}{\frac{\sqrt{{\left(-4 \cdot \left(c \cdot a\right)\right)}^{3} + {\left(b \cdot b\right)}^{3}}}{\sqrt{\left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(-4 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}}} + b}}{2}\]

  16. Final simplification0.2

    \[\leadsto \frac{\frac{c \cdot -4}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} + {\left(\left(a \cdot c\right) \cdot -4\right)}^{3}}}{\sqrt{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot -4\right)\right) + \left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\left(a \cdot c\right) \cdot -4\right)}} + b}}{2}\]

Runtime

Time bar (total: 44.1s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))